1. Field
Embodiments described herein relate to forward and backward projection using area-simulating-volume system model in three-dimensional reconstruction in a medical imaging apparatus.
2. Background
The use of positron emission tomography (PET) is growing in the field of medical imaging. In PET imaging, a radiopharmaceutical agent is introduced into an object to be imaged 15, shown in FIG. 1, via injection, inhalation, or ingestion. After administration of the radiopharmaceutical, the physical and bio-molecular properties of the agent will cause the agent to concentrate at specific locations in the human body (i.e., object 15). The actual spatial distribution of the agent, the intensity of the region of accumulation of the agent, and the kinetics of the process from administration to eventually elimination are all factors that may have clinical significance. During this process, a positron emitter attached to the radiopharmaceutical agent will emit positrons according to the physical properties of the isotope, such as half-life, branching ratio, etc.
The radionuclide emits positrons, and when an emitted positron collides with an electron, an annihilation event occurs, wherein the positron and electron are destroyed. Most of the time, an annihilation event produces two gamma photons at 511 keV traveling at substantially 180 degrees apart which are detected by a pair of crystals. By drawing a line between centers of a pair of crystals 10, i.e., the line-of-response (LOR), or drawing a polyhedron formed by connecting corresponding corners of a pair of crystals 10, i.e., tube-of-response (TOR), one can retrieve the likely location of the original disintegration. While this process will only identify a line (or tube) of possible interaction, by accumulating a large number of those lines (or tubes), and through a tomographic reconstruction process, the original distribution can be estimated. In addition to the location of the two scintillation events, if accurate timing (within few hundred picoseconds) is available, a time-of-flight (TOF) calculation can add more information regarding the likely position of the event along the line (or tube).
The above-described detection process must be repeated for a large number of annihilation events. While each imaging case must be analyzed to determine how many counts (i.e., paired events) are required to support the imaging task, current practice dictates that a typical 100-cm long, FDG (fluoro-deoxyglucose) study will need to accumulate several hundred million counts. The time required to accumulate this number of counts is determined by the injected dose of the agent and the sensitivity and counting capacity of the scanner.
Briefly, the PET reconstruction process finds the amount and the location of isotopes (unknown) in the patient from the data recorded in the PET system (known). One of the basic questions in the PET reconstruction process is to find detection probability, which represents the probability of a photon emitted from a voxel that can be detected by a given pair of crystals 10.
To address this question, a certain algorithm is designed to calculate detection probabilities aij for a line-of-response (LOR) i or tube-of-response (TOR) i and a specific voxel j. A conventional formula used in iterative Ordered Subset Expectation Maximization (OSEM) reconstruction is shown in Equation 1:
                                          f            _                    j                      k            +            1                          =                                                            f                _                            j              k                                      Q              j                                ⁢                                    ∑                              i                ∈                                  Sub                  t                                                                                                  ⁢                                                  ⁢                                                            a                  ij                                ⁢                                  Y                  i                                                                                                  ∑                                                                  j                        ′                                            =                      1                                        m                                    ⁢                                                                          ⁢                                                            a                      ij                                        ⁢                                                                  f                        _                                                                    j                        ′                                            k                                                                      +                                  R                  i                                +                                  S                  i                                                                                        (        1        )            
In Equation 1, aij is the probability of voxel j contributing to the TORi, Qj is a normalization term by summing all possible aij over the Subt, fj is the activity of voxel j, Yi represents the detected photons in TORi, Subt is the tth subset, and Ri and Si are random and scatter counts along TORi, respectively.
In Equation 1, aij can generally be divided into many components according to different physical effects, as shown in Equation 2:aij=cij×sensitivityij×resolutionij×attenuationi×TOFij× . . .   (2)where cij is the geometric probability, which is an important factor of aij, and is calculated according to the embodiments disclosed herein.
For most of the analytical calculations of cij, an implicit assumption is that radionuclei are distributed homogenously inside the voxel. Therefore, the probability cij is proportional to the intersected volume between the TOR and the voxel. If the volume of a voxel is a unit, cij can be directly represented by the intersected volume, as shown in Equation 3:
                              c          ij                =                              intersected            ⁢                                                  ⁢            volume            ⁢                                                  ⁢            between            ⁢                                                  ⁢            tube            ⁢                                                  ⁢            i            ⁢                                                  ⁢            and            ⁢                                                  ⁢            voxel            ⁢                                                  ⁢            j                                total            ⁢                                                  ⁢            volume            ⁢                                                  ⁢            of            ⁢                                                  ⁢            voxel            ⁢                                                  ⁢            j                                              (        3        )            In practice, the intersected volume is not always easy to calculate, especially for non-parallel-geometry systems such as cone-beam X-ray Computed Tomography (CT), cone-, fan- or parallel-beam Single Photon Emission Computed Tomography (SPECT), and PET.
Quantitative PET reconstruction requires a system response matrix as accurate as possible. Thus, a basic requirement is to accurately calculate the geometric probabilities. In the clinic, the speed of reconstruction is also very important. Therefore, a fast and accurate algorithm is needed to meet this requirement.